graph-theory

Is there any applicable approach to find two disjoint spanning trees of an undirected graph or to check if a certain graph has two disjoint spanning trees This is an example of Matroid union. Consider the graphic matroid where the basis are given by the spanning trees. Now the union of this matroid with itself is again a matroid. Your question is about the size of the basis of this matroid. (whether there exist a basis of size $2(|V|-1)$. The canonical algorithm for this is Matroid partitioning algorithm. There exist an algorithm which does does the following: It maintains a set of edges with

There is almost exactly the same question . But I still don't understand, how is this heuristic working and in what sequence vertexes are passed through. Also there is a picture in a book: That shows comparison of nearest-neghbor heuristic and what I believe is a closest-pair heuristic. From the picture I may assume that on the top picture, 0 point was selected first, but on the bottom picture there was selected the leftmost or the rightmost one. Because there is nothing said about first point selection (also the closest-pair heuristic doesn't do any actions in that), I may assume that any

I am trying to implement a randomly generated maze using Prim's algorithm. I want my maze to look like this: however the mazes that I am generating from my program look like this: I'm currently stuck on correctly implementing the steps highlighted in bold: Start with a grid full of walls. Pick a cell, mark it as part of the maze. Add the walls of the cell to the wall list. While there are walls in the list: **1. Pick a random wall from the list. If the cell on the opposite side isn't in the maze yet: Make the wall a passage and mark the cell on the opposite side as part of the maze.** Add the

Given a list of sets: S_1 : [ 1, 2, 3, 4 ] S_2 : [ 3, 4, 5, 6, 7 ] S_3 : [ 8, 9, 10, 11 ] S_4 : [ 1, 8, 12, 13 ] S_5 : [ 6, 7, 14, 15, 16, 17 ] What the most efficient way to merge all sets that share at least 2 elements? I suppose this is similar to a connected components problem. So the result would be: [ 1, 2, 3, 4, 5, 6, 7, 14, 15, 16, 17] (S_1 UNION S_2 UNION S_5) [ 8, 9, 10, 11 ] [ 1, 8, 12, 13 ] (S_4 shares 1 with S_1, and 8 with S_3, but not merged because they only share one element in each) The naive implementation is O(N^2), where N is the number of sets, which is unworkable for us.

Using disjoint-set data structure can easily get connected component of Graph. And, it just supports Incremental Connected Components . However, in my case, removing edge is very common so that I am looking for an algorithm or new structure can maintain Connected Components fully dynamically (including adding and removing edge) Thanks Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity (Holm, de Lichtenberg and Thorup 2001) gives an algorithm that allows an arbitrary sequence of edge insertions, deletions and connectivity